Abstract group 1728.2181: $\He_3\times \SD_{64}$

• Label: 1728.2181
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{5} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $41$
• Trans deg.: $288$
• Rank: $2$

Group information

Description:	$\He_3\times \SD_{64}$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Automorphism group:	$\ASL(2,3).C_8.C_4.C_2^4$, of order \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Nilpotency class:	$5$
Derived length:	$2$

This group is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	16	24	32	48	96
Elements	1	17	26	18	442	4	468	8	104	16	208	416	1728
Conjugacy classes	1	2	10	2	20	2	20	4	20	8	40	80	209
Divisions	1	2	5	2	10	1	10	1	5	1	5	5	48
Autjugacy classes	1	2	2	2	4	1	4	1	2	1	2	2	24

Dimension	1	2	3	4	6	8	12	16	24	32	48	96
Irr. complex chars.	36	135	8	0	30	0	0	0	0	0	0	0	209
Irr. rational chars.	4	17	0	5	4	5	1	5	1	4	1	1	48

Minimal presentations

Permutation degree:	$41$
Transitive degree:	$288$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	96
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{6}=b^{3}=c^{96}=[a,b]=[b,c]=1, c^{a}=bc^{79} \rangle$
Permutation group:	Degree $41$ $\langle(1,18,2,17)(3,20,4,19)(5,22,6,21)(7,24,8,23)(9,26,10,25)(11,28,12,27)(13,30,14,29) \!\cdots\! \rangle$
Direct product:	$\He_3$ $\, \times\, $ $\SD_{64}$
Semidirect product:	$(C_3\times C_{96})$ $\,\rtimes\,$ $C_6$	$(C_3^2\times Q_{32})$ $\,\rtimes\,$ $C_6$	$(\He_3\times Q_{32})$ $\,\rtimes\,$ $C_2$	$Q_{32}$ $\,\rtimes\,$ $(C_2\times \He_3)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{48}$ . $C_6^2$	$C_{96}$ . $(C_3\times C_6)$	$(C_8\times \He_3)$ . $D_4$	$(C_4\times \He_3)$ . $D_8$	all 24

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{6}^{2} \simeq C_{2}^{2} \times C_{3}^{2}$
Schur multiplier:	$C_{3}^{2}$
Commutator length:	$1$

Subgroups

There are 1007 subgroups in 176 conjugacy classes, 63 normal (27 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3^2\times D_{16}$
Commutator:	$G' \simeq$ $C_{48}$	$G/G' \simeq$ $C_6^2$
Frattini:	$\Phi \simeq$ $C_{48}$	$G/\Phi \simeq$ $C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $\He_3\times \SD_{64}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $\He_3\times \SD_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3^2\times D_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1728: $\He_3\times \SD_{64}$

Order 864: $\He_3\times C_{32}$, $\He_3\times Q_{32}$, $\He_3\times D_{16}$

Order 576: $C_3^2\times \SD_{64}$ x 4

Order 432: $C_{16}\times \He_3$, $Q_{16}\times \He_3$, $D_8\times \He_3$

Order 288: $C_3\times C_{96}$ x 4, $C_3^2\times Q_{32}$ x 4, $C_3^2\times D_{16}$ x 4

Order 216: $Q_8\times \He_3$, $D_4\times \He_3$, $C_8\times \He_3$

Order 192: $C_3\times \SD_{64}$ x 5

Order 144: $C_3\times C_{48}$ x 4, $C_3^2\times Q_{16}$ x 4, $C_3^2\times D_8$ x 4

Order 108: $C_4\times \He_3$ x 2, $C_2^2\times \He_3$

Order 96: $C_3\times Q_{32}$ x 5, $C_3\times D_{16}$ x 5, $C_{96}$ x 5

Order 72: $Q_8\times C_3^2$ x 4, $D_4\times C_3^2$ x 4, $C_3\times C_{24}$ x 4

Order 64: $\SD_{64}$

Order 54: $C_2\times \He_3$ x 2

Order 48: $C_3\times Q_{16}$ x 5, $C_3\times D_8$ x 5, $C_{48}$ x 5

Order 36: $C_3\times C_{12}$ x 8, $C_6^2$ x 4

Order 32: $Q_{32}$, $D_{16}$, $C_{32}$

Order 27: $\He_3$

Order 24: $C_{24}$ x 5, $C_3\times Q_8$ x 5, $C_3\times D_4$ x 5

Order 18: $C_3\times C_6$ x 8

Order 16: $Q_{16}$, $D_8$, $C_{16}$

Order 12: $C_{12}$ x 10, $C_2\times C_6$ x 5

Order 9: $C_3^2$ x 4

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$ x 10

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $\He_3\times \SD_{64}$

Order 864: $\He_3\times C_{32}$, $\He_3\times Q_{32}$, $\He_3\times D_{16}$

Order 576: $C_3^2\times \SD_{64}$

Order 432: $C_{16}\times \He_3$, $Q_{16}\times \He_3$, $D_8\times \He_3$

Order 288: $C_3\times C_{96}$, $C_3^2\times Q_{32}$, $C_3^2\times D_{16}$

Order 216: $Q_8\times \He_3$, $D_4\times \He_3$, $C_8\times \He_3$

Order 192: $C_3\times \SD_{64}$ x 2

Order 144: $C_3\times C_{48}$, $C_3^2\times Q_{16}$, $C_3^2\times D_8$

Order 108: $C_4\times \He_3$ x 2, $C_2^2\times \He_3$

Order 96: $C_3\times Q_{32}$ x 2, $C_3\times D_{16}$ x 2, $C_{96}$ x 2

Order 72: $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_3\times C_{24}$

Order 64: $\SD_{64}$

Order 54: $C_2\times \He_3$ x 2

Order 48: $C_3\times Q_{16}$ x 2, $C_3\times D_8$ x 2, $C_{48}$ x 2

Order 36: $C_3\times C_{12}$ x 2, $C_6^2$

Order 32: $Q_{32}$, $D_{16}$, $C_{32}$

Order 27: $\He_3$

Order 24: $C_{24}$ x 2, $C_3\times Q_8$ x 2, $C_3\times D_4$ x 2

Order 18: $C_3\times C_6$ x 2

Order 16: $Q_{16}$, $D_8$, $C_{16}$

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$ x 4

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $\He_3\times \SD_{64}$ ($C_1$)

Order 864: $\He_3\times C_{32}$ ($C_2$), $\He_3\times Q_{32}$ ($C_2$), $\He_3\times D_{16}$ ($C_2$)

Order 576: $C_3^2\times \SD_{64}$ ($C_3$) x 4

Order 432: $C_{16}\times \He_3$ ($C_2^2$)

Order 288: $C_3\times C_{96}$ ($C_6$) x 4, $C_3^2\times Q_{32}$ ($C_6$) x 4, $C_3^2\times D_{16}$ ($C_6$) x 4

Order 216: $C_8\times \He_3$ ($D_4$)

Order 192: $C_3\times \SD_{64}$ ($C_3^2$)

Order 144: $C_3\times C_{48}$ ($C_2\times C_6$) x 4

Order 108: $C_4\times \He_3$ ($D_8$)

Order 96: $C_3\times Q_{32}$ ($C_3\times C_6$), $C_3\times D_{16}$ ($C_3\times C_6$), $C_{96}$ ($C_3\times C_6$)

Order 72: $C_3\times C_{24}$ ($C_3\times D_4$) x 4

Order 64: $\SD_{64}$ ($\He_3$)

Order 54: $C_2\times \He_3$ ($D_{16}$)

Order 48: $C_{48}$ ($C_6^2$)

Order 36: $C_3\times C_{12}$ ($C_3\times D_8$) x 4

Order 32: $Q_{32}$ ($C_2\times \He_3$), $D_{16}$ ($C_2\times \He_3$), $C_{32}$ ($C_2\times \He_3$)

Order 27: $\He_3$ ($\SD_{64}$)

Order 24: $C_{24}$ ($D_4\times C_3^2$)

Order 18: $C_3\times C_6$ ($C_3\times D_{16}$) x 4

Order 16: $C_{16}$ ($C_2^2\times \He_3$)

Order 12: $C_{12}$ ($C_3^2\times D_8$)

Order 9: $C_3^2$ ($C_3\times \SD_{64}$) x 4

Order 8: $C_8$ ($D_4\times \He_3$)

Order 6: $C_6$ ($C_3^2\times D_{16}$)

Order 4: $C_4$ ($D_8\times \He_3$)

Order 3: $C_3$ ($C_3^2\times \SD_{64}$)

Order 2: $C_2$ ($\He_3\times D_{16}$)

Order 1: $C_1$ ($\He_3\times \SD_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $\He_3\times \SD_{64}$ ($C_1$)

Order 864: $\He_3\times C_{32}$ ($C_2$), $\He_3\times Q_{32}$ ($C_2$), $\He_3\times D_{16}$ ($C_2$)

Order 576: $C_3^2\times \SD_{64}$ ($C_3$)

Order 432: $C_{16}\times \He_3$ ($C_2^2$)

Order 288: $C_3\times C_{96}$ ($C_6$), $C_3^2\times Q_{32}$ ($C_6$), $C_3^2\times D_{16}$ ($C_6$)

Order 216: $C_8\times \He_3$ ($D_4$)

Order 192: $C_3\times \SD_{64}$ ($C_3^2$)

Order 144: $C_3\times C_{48}$ ($C_2\times C_6$)

Order 108: $C_4\times \He_3$ ($D_8$)

Order 96: $C_3\times Q_{32}$ ($C_3\times C_6$), $C_3\times D_{16}$ ($C_3\times C_6$), $C_{96}$ ($C_3\times C_6$)

Order 72: $C_3\times C_{24}$ ($C_3\times D_4$)

Order 64: $\SD_{64}$ ($\He_3$)

Order 54: $C_2\times \He_3$ ($D_{16}$)

Order 48: $C_{48}$ ($C_6^2$)

Order 36: $C_3\times C_{12}$ ($C_3\times D_8$)

Order 32: $Q_{32}$ ($C_2\times \He_3$), $D_{16}$ ($C_2\times \He_3$), $C_{32}$ ($C_2\times \He_3$)

Order 27: $\He_3$ ($\SD_{64}$)

Order 24: $C_{24}$ ($D_4\times C_3^2$)

Order 18: $C_3\times C_6$ ($C_3\times D_{16}$)

Order 16: $C_{16}$ ($C_2^2\times \He_3$)

Order 12: $C_{12}$ ($C_3^2\times D_8$)

Order 9: $C_3^2$ ($C_3\times \SD_{64}$)

Order 8: $C_8$ ($D_4\times \He_3$)

Order 6: $C_6$ ($C_3^2\times D_{16}$)

Order 4: $C_4$ ($D_8\times \He_3$)

Order 3: $C_3$ ($C_3^2\times \SD_{64}$)

Order 2: $C_2$ ($\He_3\times D_{16}$)

Order 1: $C_1$ ($\He_3\times \SD_{64}$)

Series

Derived series

$\He_3\times \SD_{64}$

$\rhd$

$C_{48}$

$\rhd$

$C_1$

Chief series

$\He_3\times \SD_{64}$

$\rhd$

$\He_3\times D_{16}$

$\rhd$

$C_{16}\times \He_3$

$\rhd$

$C_3\times C_{48}$

$\rhd$

$C_{48}$

$\rhd$

$C_{24}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$\He_3\times \SD_{64}$

$\rhd$

$C_{48}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_4\times \He_3$

$\lhd$

$C_8\times \He_3$

$\lhd$

$C_{16}\times \He_3$

$\lhd$

$\He_3\times \SD_{64}$

Character theory

Complex character table

See the $209 \times 209$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.